On arithmetic nature of q-analogue of the generalized Stieltjes constants

In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the q-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a q-analogue of the Hurwitz zeta function around s=1. We establish the closed-form expressions for the first two coefficients in the Laurent series of the q-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to ₀ (q, x) for q>1 and 0 < x <1, where ₀ (q, x) is the constant term which appears in the Laurent series expansion of q-Hurwitz zeta function around s=1. Furthermore, we put forth a conjecture about the linear independence of special values of ₀ (q, x) along with 1 at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals.